Recent approaches to controllable 4D video generation often rely on fine-tuning pre-trained Video Diffusion Models (VDMs). This dominant paradigm is computationally expensive, requiring large-scale datasets and architectural modifications, and frequently suffers from catastrophic forgetting of the model's original generative priors. Here, we propose InverseCrafter, an efficient inpainting inverse solver that reformulates the 4D generation task as an inpainting problem solved in the latent space. The core of our method is a principled mechanism to encode the pixel space degradation operator into a continuous, multi-channel latent mask, thereby bypassing the costly bottleneck of repeated VAE operations and backpropagation. InverseCrafter not only achieves comparable novel view generation and superior measurement consistency in camera control tasks with near-zero computational overhead, but also excels at general-purpose video inpainting with editing. Code is available at https://github.com/yeobinhong/InverseCrafter.